Kadeishvili's proof of the minimality theorem gives us an algorithm for computing A-infinity algebra structures on the homology of a dg-algebra which will be compatible with the original dg-algebra in all the expected ways. However, this computational route will only provide us with partial evaluations of the structure maps of the resulting A-infinity structure. In this talk, I will describe recent developments from my thesis and subsequent work, giving conditions that will allow for the reduction of computational load for constructing A-infinity structures while still retaining the same amount of information. These methods make a computer-aided study of A-infinity structures feasible and provide a criterion for detecting finite and completely computed A-infinity algebras.